A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.
YouTube Encyclopedic

1/5Views:449 21216 982438350 2148 323

✪ Example: Complex roots for a quadratic  Algebra II  Khan Academy

✪ Precalculus 12.1a  Complex Quadratic

✪ Factoring a Complex Quadratic

✪ Quadratic inequalities  Polynomial and rational functions  Algebra II  Khan Academy

✪ Quadratic equations with complex coefficients
Transcription
We're asked to solve 2x squared plus 5 is equal to 6x. And so we have a quadratic equation here. But just to put it into a form that we're more familiar with, let's try to put it into standard form. And standard form, of course, is the form ax squared plus bx plus c is equal to 0. And to do that, we essentially have to take the 6x and get rid of it from the right hand side. So we just have a 0 on the right hand side. And to do that, let's just subtract 6x from both sides of this equation. And so our left hand side becomes 2x squared minus 6x plus 5 is equal to and then on our right hand side, these two characters cancel out, and we just are left with 0. And there's many ways to solve this. We could try to factor it. And if I was trying to factor it, I would divide both sides by 2. If I divide both sides by 2, I would get integer coefficients on the x squared in the x term, but I would get 5/2 for the constant. So it's not one of these easy things to factor. We could complete the square, or we could apply the quadratic formula, which is really just a formula derived from completing the square. So let's do that in this scenario. And the quadratic formula tells us that if we have something in standard form like this, that the roots of it are going to be negative b plus or minus so that gives us two roots right over there plus or minus square root of b squared minus 4ac over 2a. So let's apply that to this situation. Negative b this right here is b. So negative b is negative negative 6. So that's going to be positive 6, plus or minus the square root of b squared. Negative 6 squared is 36, minus 4 times a which is 2 times 2 times c, which is 5. Times 5. All of that over 2 times a. a is 2. So 2 times 2 is 4. So this is going to be equal to 6 plus or minus the square root of 36 so let me just figure this out. 36 minus so this is 4 times 2 times 5. This is 40 over here. So 36 minus 40. And you already might be wondering what's going to happen here. All of that over 4. Or this is equal to 6 plus or minus the square root of negative 4. 36 minus 40 is negative 4 over 4. And you might say, hey, wait Sal. Negative 4, if I take a square root, I'm going to get an imaginary number. And you would be right. The only two roots of this quadratic equation right here are going to turn out to be complex, because when we evaluate this, we're going to get an imaginary number. So we're essentially going to get two complex numbers when we take the positive and negative version of this root. So let's do that. So the square root of negative 4, that is the same thing as 2i. And we know that's the same thing as 2i, or if you want to think of it this way. Square root of negative 4 is the same thing as the square root of negative 1 times the square root of 4, which is the same. I could even do it one step that's the same thing as negative 1 times 4 under the radical, which is the same thing as the square root of negative 1 times the square root of 4. And the principal square root of negative 1 is i times the principal square root of 4 is 2. So this is 2i, or i times 2. So this right over here is going to be 2i. So we are left with x is equal to 6 plus or minus 2i over 4. And if we were to simplify it, we could divide the numerator and the denominator by 2. And so that would be the same thing as 3 plus or minus i over 2. Or if you want to write them as two distinct complex numbers, you could write this as 3 plus i over 2, or 3/2 plus 1/2i. That's if I take the positive version of the i there. Or we could view this as 3/2 minus 1/2i. This and these two guys right here are equivalent. Those are the two roots. Now what I want to do is a verify that these work. Verify these two roots. So this one I can rewrite as 3 plus i over 2. These are equivalent. All I did you can see that this is just dividing both of these by 2. Or if you were to essentially factor out the 1/2, you could go either way on this expression. And this one over here is going to be 3 minus i over 2. Or you could go directly from this. This is 3 plus or minus i over 2. So 3 plus i over 2. Or 3 minus i over 2. This and this or this and this, or this. These are all equal representations of both of the roots. But let's see if they work. So I'm first going to try this character right over here. It's going to get a little bit hairy, because we're going to have to square it and all the rest. But let's see if we can do it. So what we want to do is we want to take 2 times this quantity squared. So 2 times 3 plus i over 2 squared plus 5. And we want to verify that that's the same thing as 6 times this quantity, as 6 times 3 plus i over 2. So what is 3 plus i squared? So this is 2 times let me just square this. So 3 plus i, that's going to be 3 squared, which is 9, plus 2 times the product of three and i. So 3 times i is 3i, times 2 is 6i. So plus 6i. And if that doesn't make sense to you, I encourage you to kind of multiply it out either with the distributive property or FOIL it out, and you'll get the middle term. You'll get 3i twice. When you add them, you get 6i. I And then plus i squared, and i squared is negative 1. Minus 1. All of that over 4, plus 5, is equal to well, if you divide the numerator and the denominator by 2, you get a 3 here and you get a 1 here. And 3 distributed on 3 plus i is equal to 9 plus 3i. And what we have over here, we can simplify it just to save some screen real estate. 9 minus 1 is 8. So if I get rid of this, this is just 8 plus 6i. We can divide the numerator and the denominator right here by 2. So the numerator would become 4 plus 3i, if we divided it by 2, and the denominator here is just going to be 2. This 2 and this 2 are going to cancel out. So on the left hand side, we're left with 4 plus 3i plus 5. And this needs to be equal to 9 plus 3i. Well, you can see we have a 3i on both sides of this equation. And we have a 4 plus 5, which is exactly equal to 9. So this solution, 3 plus i, definitely works. Now let's try 3 minus i. So once again, just looking at the original equation, 2x squared plus 5 is equal to 6x. Let me write it down over here. Let me rewrite the original equation. We have 2x squared plus 5 is equal to 6x. And now we're going to try this root, verify that it works. So we have 2 times 3 minus i over 2 squared plus 5 needs to be equal to 6 times this business. 6 times 3 minus i over 2. Once again, a little hairy. But as long as we do everything, we put our head down and focus on it, we should be able to get the right result. So 3 minus i squared. 3 minus i times 3 minus i, which is and you could get practice taking squares of two termed expressions, or complex numbers in this case actually it's going to be 9, that's 3 squared, and then 3 times negative i is negative 3i. And then you're going to have two of those. So negative 6i. So negative i squared is also negative 1. That's negative 1 times negative 1 times i times i. So that's also negative 1. Negative i squared is also equal to negative 1. Negative i is also another square root. Not the principal square root, but one of the square roots of negative 1. So now we're going to have a plus 1, because oh, sorry, we're going to have a minus 1. Because this is negative i squared, which is negative 1. And all of that over 4. All of that over that's 2 squared is 4. Times 2 over here, plus 5, needs to be equal to well, before I even multiply it out, we could divide the numerator and the denominator by 2. So 6 divided by 2 is 3. 2 divided by 2 is 1. So 3 times 3 is 9. 3 times negative i is negative 3i. And if we simplify it a little bit more, 9 minus 1 is going to be I'll do this in blue. 9 minus 1 is going to be 8. We have 8 minus 6i. And then if we divide 8 minus 6i by 2 and 4 by 2, in the numerator, we're going to get 4 minus 3i. And in the denominator over here, we're going to get a 2. We divided the numerator and the denominator by 2. Then we have a 2 out here. And we have a 2 in the denominator. Those two characters will cancel out. And so this expression right over here cancels or simplifies to 4 minus 3i. Then we have a plus 5 needs to be equal to 9 minus 3i. I We have a negative 3i on the left, a negative 3i on the right. We have a 4 plus 5. We could evaluate it. This left hand side is 9 minus 3i, which is the exact same complex number as we have on the right hand side, 9 minus 3i. So it also checks out. It is also a root. So we verified that both of these complex roots, satisfy this quadratic equation.
Contents
Properties
Quadratic polynomials have the following properties, regardless of the form:
 It is an unicritical polynomial, i.e. it has one critical point,
 It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic.^{[1]}
 It is a unimodal function,
 It is a rational function,
 It is an entire function.
Forms
When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms:
 The general form: where
 The factored form used for logistic map
 which has an indifferent fixed point with multiplier at the origin^{[2]}
 The monic and centered form,
The monic and centered form has been studied extensively, and has the following properties:
 It is the simplest form of a nonlinear function with one coefficient (parameter),
 It is a centered polynomial (the sum of its critical points is zero).^{[3]}
The lambda form is:
 the simplest nontrival perturbation of unperturbated system
 "the first family of dynamical systems in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"^{[4]}
Conjugation
Between forms
Since is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia and Fatou sets.
When one wants change from to :^{[5]}
When one wants change from to the parameter transformation is^{[6]}
and the transformation between the variables in and is
With doubling map
There is semiconjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.
Notation
Iteration
Here denotes the nth iteration of the function (and not exponentiation of the function):
so
Because of the possible confusion with exponentiation, some authors write for the nth iterate of the function
Parameter
The monic and centered form can be marked by:
 the parameter
 the external angle of the ray that lands:
 at c in M on the parameter plane
 at z = c in J(f) on the dynamic plane
so :
Map
The monic and centered form, sometimes called the DouadyHubbard family of quadratic polynomials,^{[7]} is typically used with variable and parameter :
When it is used as an evolution function of the discrete nonlinear dynamical system
it is named the quadratic map:^{[8]}
The Mandelbrot set is the set of values of the parameter c for which the initial condition z_{0} = 0 does not cause the iterates to diverge to infinity.
Critical items
Critical point
A critical point of is a point in the dynamical plane such that the derivative vanishes:
Since
implies
we see that the only (finite) critical point of is the point .
is an initial point for Mandelbrot set iteration.^{[9]}
Critical value
A critical value of is the image of a critical point:
Since
we have
So the parameter is the critical value of
Critical orbit
The forward orbit of a critical point is called a critical orbit. Critical orbits are very important because every attracting periodic orbit attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.^{[10]}^{[11]}^{[12]}
This orbit falls into an attracting periodic cycle if one exists.
Wikimedia Commons has media related to Category:Critical orbits. 
Critical sector
The critical sector is a sector of the dynamical plane containing the critical point.
Critical polynomial
so
These polynomials are used for:
 finding centers of these Mandelbrot set components of period n. Centers are roots of nth critical polynomials
 finding roots of Mandelbrot set components of period n (local minimum of )
 Misiurewicz points
Critical curves
Diagrams of critical polynomials are called critical curves.^{[13]}
These curves create the skeleton (the dark lines) of a bifurcation diagram.^{[14]}^{[15]}
Spaces, planes
4D space
One can use the JuliaMandelbrot 4dimensional ( 4D) space for a global analysis of this dynamical system.^{[16]}
In this space there are 2 basic types of 2D planes:
 the dynamical (dynamic) plane, plane or cplane
 the parameter plane or zplane
There is also another plane used to analyze such dynamical systems wplane:
 the conjugation plane^{[17]}
 model plane^{[18]}
2D Parameter plane
The phase space of a quadratic map is called its parameter plane. Here:
is constant and is variable.
There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
The parameter plane consists of:
 The Mandelbrot set
 The bifurcation locus = boundary of Mandelbrot set with
 root points
 Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set^{[19]} with internal rays
 The bifurcation locus = boundary of Mandelbrot set with
 exterior of Mandelbrot set with
 external rays
 equipotential lines
There are many different subtypes of the parameter plane.^{[20]}^{[21]}
See also :
 Boettcher map which maps exterior of mandelbrot set to the exterior of unit disc
 multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc
2D Dynamical plane
"The polynomial Pc maps each dynamical ray to another ray doubling the angle (which we measure in full turns, i.e. 0 = 1 = 2π rad = 360◦), and the dynamical rays of any polynomial “look like straight rays” near infinity. This allows us to study the Mandelbrot and Julia sets combinatorially, replacing the dynamical plane by the unit circle, rays by angles, and the quadratic polynomial by the doubling modulo one map." Virpi K a u k o^{[22]}
On the dynamical plane one can find:
 The Julia set
 The Filled Julia set
 The Fatou set
 Orbits
The dynamical plane consists of:
Here, is a constant and is a variable.
The twodimensional dynamical plane can be treated as a Poincaré crosssection of threedimensional space of continuous dynamical system.^{[23]}^{[24]}
Dynamical zplanes can be divided in two groups :
 plane for ( see complex squaring map )
 planes ( all other planes for )
Riemann sphere
The extended complex plane plus a point at infinity
Derivatives
First derivative with respect to c
On the parameter plane:
 is a variable
 is constant
The first derivative of with respect to c is
This derivative can be found by iteration starting with
and then replacing at every consecutive step
This can easily be verified by using the chain rule for the derivative.
This derivative is used in the distance estimation method for drawing a Mandelbrot set.
First derivative with respect to z
On the dynamical plane:
 is a variable;
 is a constant.
At a fixed point
At a periodic point z_{0} of period p the first derivative of a function
is often represented by and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. It used to check the stability of periodic (also fixed) points.
At a nonperiodic point, the derivative, denoted by can be found by iteration starting with
and then using
This derivative is used for computing the external distance to the Julia set.
Schwarzian derivative
The Schwarzian derivative (SD for short) of f is:^{[25]}
 .
See also
 Misiurewicz point
 Periodic points of complex quadratic mappings
 Mandelbrot set
 Julia set
 Milnor–Thurston kneading theory
 Tent map
 Logistic map
References
 ^ Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits
 ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
 ^ Bodil Branner: Holomorphic dynamical systems in the complex plane. MatReport No 199642. Technical University of Denmark
 ^ Dynamical Systems and Small Divisors, Editors: Stefano Marmi, JeanChristophe Yoccoz, page 46
 ^ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
 ^ stackexchange questions : Show that the familiar logistic map ...
 ^ Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236264
 ^ Weisstein, Eric W. "Quadratic Map." From MathWorldA Wolfram Web Resource
 ^ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Archived 26 April 2012 at the Wayback Machine
 ^ M. Romera Archived 22 June 2008 at the Wayback Machine, G. Pastor Archived 1 May 2008 at the Wayback Machine, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Archived 11 December 2009 at the Wayback Machine Fractalia Archived 19 September 2008 at the Wayback Machine 6, No. 21, 1012 (1997)
 ^ Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104116
 ^ Khan Academy : Mandelbrot Spirals 2
 ^ The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640653
 ^ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971506823. Archived from the original on 5 December 2009. Retrieved 2 December 2009.
 ^ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in onedimensional quadratic maps", Physica A, 232 (1996), 517535. Preprint Archived 2 October 2006 at the Wayback Machine
 ^ JuliaMandelbrot Space at MuENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
 ^ Carleson, Lennart, Gamelin, Theodore W.: Complex Dynamics Series: Universitext, Subseries: Universitext: Tracts in Mathematics, 1st ed. 1993. Corr. 2nd printing, 1996, IX, 192 p. 28 illus., ISBN 9780387979427
 ^ Holomorphic motions and puzzels by P Roesch
 ^ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
 ^ Alternate Parameter Planes by David E. Joyce
 ^ exponentialmap by Robert Munafo
 ^ Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000)
 ^ Mandelbrot set by Saratov group of theoretical nonlinear dynamics
 ^ Moehlis, Kresimir Josic, Eric T. SheaBrown (2006) Periodic orbit. Scholarpedia,
 ^ The Schwarzian Derivative & the Critical Orbit by Wes McKinney 18.091 20 April 2005
External links
 M. Nevins and D. Rogers, "Quadratic maps as dynamical systems on the padic numbers"
 Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
Wikimedia Commons has media related to Category:Complex quadratic map. 